Wiles won 6 million Norwegian Kroner as part of the prize,
equivalent to about $700,000.

Fermat's Last Theorem was originally suggested over 350 years
ago, but Wiles' proof of the theorem involved proving a more general
result in modern algebraic geometry, bringing in complex
mathematical techniques that were not developed until the 20th
century.

Despite the complexity of the proof, the theorem itself is pretty
straightforward. Here's the fascinating problem that
Wiles solved that led to his prize.

Perfect squares and cubes

Perfect
squares are whole numbers that are the square of some other
whole number. 25 is 52, 64 is 82, and 81 is
92.

It turns out that some perfect squares are in turn the sum of two
other perfect squares: 25 = 9 + 16, or 32 +
42. Indeed, there are infinitely many such perfect
squares, and a perfect square along with the two other squares
that add up to it are called Pythagorean triples for
their geometric relationship to the
Pythagorean Theorem regarding right triangles.

Of course, not all perfect squares can be written as the sum of
two other perfect squares — a few minutes of playing around with,
say, all the ways to add together two whole numbers to get 9 will
show that there's no pair of perfect squares in that list.

When mathematicians see a phenomenon like Pythagorean triples
they often jump pretty quickly to considering generalizations of
that phenomenon. In our case, we want to know if we can extend
the idea of Pythagorean triples to higher powers: Could I take a
perfect
cube, like 8 (which equals 23 or 2 x 2 x 2), and
write it as a sum of two other perfect cubes? What about a
perfect fourth power?

Formally speaking, for a power n bigger than 2, can I find whole
numbers a, b, and c so that cn = an +
bn?

Wiles won the Abel prize for his 1994 proof that the answer
to that question is an emphatic "no": Triples like this don't
exist for powers bigger than 2.

Given that this is, on the surface, a fairly simple question,
it's remarkable that the answer wasn't proven for over 350
years since it was first posited.

Fermat's Last Theorem

This mathematical statement — that we can't take perfect whole
number powers and break them into sums of other perfect powers —
is known as Fermat's Last Theorem after the 17th century French
lawyer and mathematician
Pierre de Fermat.

The theorem is named after Fermat because of an amazing note
written in the margin of his copy of a classic Greek math text
around 1637,
as per Wikipedia (emphasis ours):

"It is impossible to separate a cube into two cubes, or a
fourth power into two fourth powers, or in general, any power
higher than the second, into two like powers. I have
discovered a truly marvellous proof of this, which this margin is
too narrow to contain."

Of course, Fermat never actually published this supposed
"truly marvellous proof." Given that Wiles' proof of the theorem
used elaborate mathematical techniques that didn't exist until
the second half of the 20th century, many contemporary
mathematicians are skeptical that Fermat had an actual
proof.

This is not to say that the estimable lawyer was trying to
commit fraud. It's far from uncommon for a mathematician to
convince him or herself that they've cracked a problem only to
later realize (or for their peers to realize and inform them)
that they missed some key, subtle detail or another that derails
the whole project.

Regardless of whether or not Fermat actually had a proof,
his method of claiming a solution that he didn't write down
because he didn't have enough space has definitely tempted
many math students confronting homework deadlines in the
centuries since.